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UCLA Department of Economics
Fall 2013
PhD. Qualifying Exam in Macroeconomic Theory
Instructions: This exam consists of three parts, and you are to complete each part.
Answer each part in a separate bluebook. All three parts will receive equal
weight in your grade.
Part I
Consider a problem solved by a planner who maximizes the following:

E   t U (ct ,1  ht ).
t 0
Here, ct is consumption, ht is hours worked, the function U is assumed to have all the usual
properties, and 0 <  < 1. Output is produced according to a constant returns to scale
technology, yt  zt F (kt , ht ) , where yt is output and kt is the stock of capital. The variable zt is a
technology shock that evolves through time according to a first order autoregressive process with
an unconditional mean of 1 and unconditional variance of  z2 . The stock of capital is assumed to
depreciate at the rate  each period.
Output can be used for consumption, investment ( it ) or government purchases ( gt ).
Investment in period t becomes productive capital one period later, kt 1  (1   )kt  it .
Government spending is an exogenous random variable that, like the technology shock, follows a
first order autoregressive process, in this case with an unconditional mean of g and
unconditional variance of  g2 . Innovations to this process are assumed to be independent of
innovations to the technology shock process. In addition, government purchases are financed
with lump sum taxes. Assume, initially, that government purchases do not directly affect
preferences or the technology; they are simply thrown into the sea.
(a) Suggest functional forms for the utility function and the production function that are
consistent with balanced growth properties. Defend your choices.
(b) Specify first order autoregressive processes for zt and gt that have the desired
unconditional means and variances.
(c) Formulate the planner’s problem as a dynamic programming problem.
(d) It has been shown that for a calibrated version of the above model without government
spending, the contemporaneous correlation between hours and productivity is close to
one. Once stochastic government spending is added, the correlation becomes lower.
Provide intuition for this finding.
(e) Formulate the problem that would be solved by a typical household and firm in a
decentralized version of this economy. Define a recursive competitive equilibrium for
this economy that includes a government that collects lump sum taxes and disposes of the
proceeds.
2
(f) Is the competitive equilibrium you have defined equivalent to the allocation that would
be chosen by a social planner? You do not need to provide a rigorous proof of this, but
state your answer and provide some explanation. Suggest a modified version of this
economy where the opposite result would hold.
(g) Now suppose that government expenditures are used to purchase consumption goods that
are perfect substitutes for ct in an agent’s utility function. Specify the social planning
problem for such an economy as a dynamic programming problem. Do you expect that
the result described in part (d) would hold for this model? Explain.
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Part II
1
Monetary Policy and Exchange Rates in a Growth
Model
There are two consumption goods and two money stocks in the following perfect
foresight, in…nite horizon economy. Upper case letters are per-capita variables,
and lower-case letters are household variables.
Preferences for the represenative household are given by
X
t
f
c~1
1
+ v(lt )g
c~ = [ ca + (1
)cb ]
1
The individual time constraint is
1
lt + hat + hbt
The economy’s resource constraint is
1
Aa Kat Hat
Cat
1
Ab Kbt Hbt
Cbt + It
It
= Kt+1
(1
)Kt
Capital and labor can be costlessly reallocated across sectors.
There are two cash-in-advance constraints, one for good ca and one for good
cb : The household CIA constraints are:
ma
pa ca
mb
pb cb
Monetary policy for ma and mb is exogenous and given by
Mat+1
=
at Mat
Mbt+1
=
bt Mbt
Monetary injections or withdrawls are implemented via lump-sum transfers
1
The per-capita endowments of ma ; mb ; and k at date 0 are given.
(1) What restriction(s) do you need to place on the function v? What
restriction do you need to place on the parameter ; and what features of the
model do and govern? For preferences over the consumption aggregate c~;
what restriction do you need to place on ? How do you need to modify the
problem for the case of ! 0?
(2) Write this problem as a competitive equilibrium. Solve for the …rst-order
conditions. De…ne the object q as the exchange rate between the two moneys,
ma and mb : Discuss how model parameters and state variables determine this
exchange rate. Describe how this exchange rate would change over time as a
function of a and b :
(3) Suppose that there was labor-augmenting technical progress in the two
production functions, in which technological growth occured at a constant rate.
Describe how you would construct a stationary equilibrium for this economy.
(You can write down the equations if you prefer, otherwise, just state in words
what you would do).
(4) Consider a steady state of this economy. Under what set of monetary
policies does the competitive equilibrium coincide with the social optimum?
(5) Suppose that at = $ > 1 for all t, and consider a steady state. Under
what monetary policies is this economy equivalent to the cash good - credit
good model of Lucas and Stokey? (Recall in Lucas-Stokey that there are multiple consumption goods, some of which require cash for purchase, while other
consumptions goods and investment dont require cash).
(6) Consider a steady state, in which there are positive and constant rates
of money growth for both currencies. Under these conditions, show that this
economy is equivalent to the Lucas -Stokey model in which there is a speci…c
tax and transfer policy in the Lucas-Stokey economy.
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Part III
Human Capital and the Duration of Unemployment:
One of the puzzling features of the labor market in this recession is that there is
a large stock of unemployed workers who have been unemployed for a long time.
We explore how this might happen in a Mortensen-Pissarides search model with
depreciation of skills during unemployment.
Time is continuous and labelled t ≥ 0. There is a measure one of agents in the
labor force (who can work). These agents in the labor force can be of one of two
types: high productivity and low productivity. High productivity agents produce
output yH per unit time that they are employed in a match. Low productivity
agents produce output yL < yH per unit time that they are employed in a match.
These agents can also have one of two employment statuses: they can be employed in a match or unemployed. We assume that unemployed workers consume b
per unit time while unemployed and that yH > b. We will consider cases in which
yL > b and yL < b.
At each date t, let lHt denote the measure of agents who are employed in matches
and have high productivity, lLt the measure of agents who are employed in matches
and have low productivity, uHt , the measure of agents who are unemployed and
have high productivity, and uLt the measure of workers who are unemployed and
have low productivity. At each moment in time, we have lHt + lLt + uHt + uLt = 1.
Agents gain skills when they are matched in a job and lose skills when they
are unemployed. Agents who are employed in a match also lose their jobs (their
matches are destroyed) at an exogenous rate independent of their skill level and
agents who are unemployed become employed at an endogenous rate that does
depend on their skill level in a manner to be described below.
Specifically, assume that in steady-state employed agents lose their jobs at rate
λ > 0 per unit time and that high skilled unemployed agents enter into a new match
at rate αuH > 0 per unit time while low skilled unemployed agents enter into a new
match at rate αuL ≥ 0 per unit time. Low productivity agents who are employed
in a match become high productivity agents at rate η per unit time (representing
learning on the job) and high productivity agents who are unemployed become low
productivity agents at rate δ per unit time (representing loss of human capital in
unemployment).
Part A: Take the transition rates λ, αuH , αuL , η, and δ as parameters. Compute
the steady state levels of lH , lL , uH and uL .
Consider two cases in your solution. In the first case, αuL > 0. That is, low
skilled unemployed agents find jobs. In the second case, αuL = 0. That is, low
skilled unemployed agents do not find jobs.
Give some intuition based on the parameters yH , yL and b as to why we might
have αuL = 0 in an equilibrium.
1
2
Which parameter controls the duration of unemployment for low skilled unemployed agents? Show that the level of unemployment of low skilled agents (uL ) is
increasing as the duration of unemployment for low skilled agents increases.
End of Part A
To hire unemployed agents, firms post vacancies. Firms can identify low and
high skilled unemployed agents and thus can post separate vacancies for each
type of worker. Let vHt denote the measure of vacancies posted at date t for
high skilled unemployed agents and vLt denote the measure of vacancies posted
at date t for low skilled unemployed agents. The measure of matches between
unemployed agents and vacancies for each type of agent at t is given by m(uJt , vJt )
for J ∈ {H, L} with the standard properties (non-negative, increasing and strictly
concave in each argument, m(0, v) = m(u, 0) = 0, and m(u.v) constant returns
to scale). We assume that each unemployed agent of each type finds a match at
t at rate αuJt = m(uJt , vJt )/uJt for J ∈ {H, L} and each firm with a vacancy of
type J finds a match at t at rate αeJt = m(uJt , vJt )/vJt . From here forward, we
will consider steady-states of this economy in which uJt , vJt and the corresponding
rates αuJt and αeJt are constant over time.
Firms that have posted a vacancy must pay k > 0 per unit time to post that
vacancy. The productivity of a match of type J is yJ as described above.
Let WH (wH ) denote the value to a high productivity worker of being employed
at wage wH and let UH denote the value to that worker of being unemployed. Let
JH (yH − wH ) denote the value to a firm of having a high productivity worker with
productivity yH hired at wage wH and VH denote the value of an unfilled vacancy
for a high productivity worker.
Let WL (wL ) denote the value to a low productivity worker of being employed
at wage wL and let UL denote the value to that agent of being unemployed. Let
JL (yL − wL ) denote the value to a firm of having a low productivity worker with
productivity yL hired at wage wL and VL denote the value of an unfilled vacancy
for a low productivity worker.
Assume that all agents and firms discount the future at rate r > 0. Assume
that a low productivity agent in a match who switches to being high productivity
renegotiates his wage as a high productivity worker when the switch occurs.
Part B: Write the Bellman Equations for the firm and the worker defining
WJ (w), JJ (yJ − w), UJ , and VJ for J ∈ {H, L}. Take care to consider the fact
that agents can change productivities.
Assume that there is free entry into creating vacancies, so that VH = VL = 0.
Also assume that low and high productivity workers have the same bargaining
power. Use this condition and your Bellman equations to argue that with yL < yH ,
then we must have αeL > αeH . Use the fact that the matching function has
constant returns to scale to argue that αuL < αuH .